Potts models with invisible states on general Bethe lattices

The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q = 2 case, a Bragg-Williams (mean-field) approach necessitates four such invisible states while a 3-regular random graph formalism requires seventeen. In both of these cases, the changeover from second-to first-order behaviour induced by the invisible states is identified through the tricritical point of an equivalent Blume-Emery-Griffiths model. Here we investigate the generalized Potts model on a Bethe lattice with z neighbours. We show that, in the q = 2 case, r(c)(z) = 4z/3(z-1) (z-1/z-2)(z) invisible states are required to manifest the equivalent Blume-Emery-Griffiths tricriticality. When z = 3, the 3-regular random graph result is recovered, while z ->infinity delivers the Bragg-Williams (mean-field) result.